Problem: $\int x^2(2x-5)\,dx=$ $+C$
Answer: The integrand is the product of two functions: $x^2$ and $2x-5$. Although it is tempting to take the product of their integrals, this would not work. $\int f(x)\cdot g(x)\,dx\neq\int f(x)\,dx \cdot \int g(x)\,dx$ Instead, what we should do is expand the parentheses so we get a nice polynomial. $\int x^2(2x-5)\,dx=\int (2x^3-5x^2)\,dx$ Now we can integrate using the reverse power rule, the sum rule, and the constant multiple rule for indefinite integrals. $\begin{aligned} &\phantom{=}\int x^2(2x-5)\,dx \\\\ &=\int (2x^3-5x^2)\,dx \\\\ &= 2\int x^3\,dx-5\int x^2\,dx \\\\ &=2\dfrac{x^4}{4}-5\dfrac{x^3}{3}+C \\\\ &=\dfrac12x^4-\dfrac53x^3+C \end{aligned}$ In conclusion, $\int x^2(2x-5)\,dx=\dfrac12x^4-\dfrac53x^3+C$